Related papers: Uncertainty relations on nilpotent Lie groups

Uncertainty relations provide constraints on how well the outcomes of incompatible measurements can be predicted, and, as well as being fundamental to our understanding of quantum theory, they have practical applications such as for…

Let (H_t) be the Ornstein-Uhlenbeck semigroup on R^d with covariance matrix I and drift matrix \lambda(R-I), where \lambda>0 and R is a skew-adjoint matrix and denote by \gamma_\infty the invariant measure for (H_t). Semigroups of this form…

Let $\mathbb G$ be a step-two nilpotent group of H-type with Lie algebra $\mathfrak G=V\oplus \mathfrak t$. We define a class of vector fields $X=\{X_j\}$ on $\mathbb G$ depending on a real parameter $k\ge 1$, and we consider the…

Sharp uncertainty relations restricting the values of variances in the position space and in the momentum (wavevector) space are derived. They have the same form $\Delta r\Delta k\ge 5/2$ in the classical theory of light beams, in the…

Heisenberg's uncertainty relation can be written in terms of the step-up and step-down operators in the harmonic oscillator representation. It is noted that the single-variable Heisenberg commutation relation contains the symmetry of the…

We survey some aspects of the pseudo-differential Weyl calculus for irreducible unitary representations of nilpotent Lie groups, ranging from the classical ideas to recently obtained results. The classical Weyl-H\"ormander calculus is…

We give some estimates of the remainder terms for several conformally-invariant Sobolev-type inequalities on the Heisenberg group, in analogy with the Euclidean case. By considering the variation of associated functionals, we give a…

Quantum symmetries that leave invariant physical transition probabilities are described by projective representations of Lie groups. The mathematical theory of projected representations for topologically connected Lie groups is reviewed and…

We present the uncertainty relation for the characteristic functions (ChUR) of the quantum mechanical position and momentum probability distributions. This inequality is more general than the Heisenberg Uncertainty Relation, and is…

We prove that the Heisenberg groups can be distinguished from the other connected and simply connected Lie groups via their group $C^*$-algebras. The main step of the proof is a characterization of the nilpotent Lie groups among the…

The Hilbert space of the unitary irreducible representations of a Lie group that is a quantum dynamical group are identified with the quantum state space. Hermitian representation of the algebra are observables. The eigenvalue equations for…

We consider the uncertainty relation between position and momentum of a particle on $ S^1 $ (a circle). Since $ S^1 $ is compact, the uncertainty of position must be bounded. Consideration on the uncertainty of position demands delicate…

Results of Liflyand and collaborators on the boundedness of Hausdorff operators on the Hardy space $H^1$ over finite-dimensional real space generalized to the case of locally compact groups that are spaces of homogeneous type. Special cases…

Learning physical properties of a quantum system is essential for the developments of quantum technologies. However, Heisenberg's uncertainty principle constrains the potential knowledge one can simultaneously have about a system in quantum…

A symmetry in quantum mechanics is described by the projective representations of a Lie symmetry group that transforms between physical quantum states such that the square of the modulus of the states is invariant. The Heisenberg…

We prove a double-inequality for the product of uncertainties for position and momentum of bound states for 1D quantum mechanical systems in the semiclassical limit.

These notes describe some links between the group $\mathrm{SL}_2(\mathbb{R})$, the Heisenberg group and hypercomplex numbers---complex, dual and double numbers. Relations between quantum and classical mechanics are clarified in this…

In this paper we obtain weighted higher order Rellich, weighted Gagliardo-Nirenberg, Trudinger, Caffarelli-Kohn-Nirenberg inequalities and the uncertainty principle for Dunkl operators. Moreover, we introduce an extension of the classical…

For a simple set of observables we can express, in terms of transition probabilities alone, the Heisenberg Uncertainty Relations, so that they are proven to be not only necessary, but sufficient too, in order for the given observables to…

We present a general framework and procedure to derive uncertainty relations for observables of quantum systems in a covariant manner. All such relations are consequences of the positive semidefiniteness of the density matrix of a general…